Linear approximation ?Write an equation of the line that represents the linear approximation To the following functions at a. Then? (?a)? ?graph the function and the linear approximation near a:? (?b?) ?use the linear approximation to estimate the given quantity; and? (?c?) ?compute the percent error in your approximation. f?? ?)=?ex; a? = 0? 0.03

Solution 43E STEP 1 .The linear approximation formula is given by y = L(x) = f(a)+f (a)(xa) x Given f(x) = e and a = 0. 0 f(a) = e = 1 Then f (x) = ex f(a) = e = 1 Therefore y = L(x) = 1+1(x0) = 1+x Thus we get L(x) = 1+x Thus we get the equation of the line as y = (1+x) STEP 2 (a).graph the function and the linear approximation near a: STEP 3 (b). use the linear approximation to estimate the given quantity; 0.03 We have to estimate the value of e using the linear approximation Thus we have e0.03 L(0.03) = 10.03 = 0.97 Thus e 0.03 0.97 STEP 4 (c) .compute the percent error in your approximation. The exact value of e 0.03= 0.9704 Therefore percent error = 100× |approxexac= 100× |0.970.0.970= 100×0.00041 =0.041% |exact| |0.9704| percent error =0.041%